Department of Mathematics
 Search | Help | Login | pdf version | printable version

Math @ Duke



Publications [#244166] of Stephanos Venakides

Papers Published

  1. Shipman, SP; Venakides, S, An exactly solvable model for nonlinear resonant scattering, Nonlinearity, vol. 25 no. 9 (2012), pp. 2473-2501, ISSN 0951-7715 (doi:10.1088/0951-7715/25/9/2473.) [doi]
    (last updated on 2018/02/18)

    This work analyses the effects of cubic nonlinearities on certain resonant scattering anomalies associated with the dissolution of an embedded eigenvalue of a linear scattering system. These sharp peak-dip anomalies in the frequency domain are often called Fano resonances. We study a simple model that incorporates the essential features of this kind of resonance. It features a linear scatterer attached to a transmission line with a point-mass defect and coupled to a nonlinear oscillator. We prove two power laws in the small coupling (γ→0) and small nonlinearity (μ→0) regime. The asymptotic relation μ→Cγ 4 characterizes the emergence of a small frequency interval of triple harmonic solutions near the resonant frequency of the oscillator. As the nonlinearity grows or the coupling diminishes, this interval widens and, at the relation μ→Cγ 2, merges with another evolving frequency interval of triple harmonic solutions that extends to infinity. Our model allows rigorous computation of stability in the small μ and γ limit. The regime of triple harmonic solutions exhibits bistability - those solutions with largest and smallest response of the oscillator are linearly stable and the solution with intermediate response is unstable. © 2012 IOP Publishing Ltd & London Mathematical Society.
ph: 919.660.2800
fax: 919.660.2821

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320